The N
50Logic
Jos´e Arrazola1, Mauricio Osorio2, and Eduardo Ariza1
1Benem´erita Universidad Aut´onoma de Puebla aveariza@hotmail.com, arrazola@fcfm.buap.mx
.
2Universidad de las Am´ericas - Puebla osoriomauri@gmail.com
Abstract. We introduce a new 5-valued logic that we callN50. This logic extends GLukG, a paraconsistent logic recently introduced [10]. We show thatN50 is sound with respect to GLukG.
1 Introduction
GLukG paraconsistent logic has been investigated for a short period of time [5,10]. However, it has demonstrated important qualities for the study of other logics. It also has relevant attributes in knowledge representation [11]. Although, this work does not address this last point. On this paper, we investigate how to add a strong negation connective to GLukG logic.
The logic that is of our interest is a 5-valued logic that is sound with respect to GLukG. In this case we use two kinds of negations, one is the strong negation represented by the ∼ symbol and the other one is the default negation repre- sented by the¬symbol. In the literature we can find several logical extension, through a strong negation operator, but few of them are hardly paraconsistent logics satisfy a theorem of substitution, to see [3]. To summarize, the main contri- bution of the paper is the proposal of the paraconsistent 5-valued logic calledN50 that satisfies the following suitable properties: (1) It is a conservative extension of GLukG logic. (2) It satisfies the substitution theorem.
The structure of our paper is as follows. Section 2 describes the general back- ground needed for the paper including the definition of GLukG logic. On Section 3 we present a Hilbert-style axiomatization for GLukG which is a slight varia- tion of the one presented in [10]. On Section 4 we present Theorem 4, our main result, which establishes that our logicN50 is a conservative extension of Nelson’s N5. Finally, on Section 5 we present our conclusions and we address the future work.
We assume that the reader has some familiarity with basic logic such as chapter one in [9].
2 Background
We first introduce the syntax of logic formulas considered in this paper. Then we present a few basic definitions of how logics can be built to interpret the
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meaning of such formulas in order to, finally, give a brief introduction to several of the logics that are relevant for the results of our later sections.
2.1 Syntax of formulas
We consider a formal (propositional) language built from: an enumerable set L of elements called atoms (denoted a, b, c, . . . ); the binary connectives ∧ (con- junction), ∨ (disjunction) and → (implication); and the unary connective ¬ (negation). Formulas (denotedα,β, γ, . . . ) are constructed as usual by combi- ning these basic connectives together with the help of parentheses.
We also useα↔β to abbreviate (α→β)∧(β→α) andα←β to abbreviate β→α. It is useful to agree on some conventions to avoid the use of so many parenthesis in writing formulas. This will make the reading of complicated ex- pressions easier. First, we may omit the outer pair of parenthesis of a formula.
Second, the connectives are ordered as follows: ¬,∧,∨,→, and↔, and parent- heses are eliminated according to the rule that, first, ¬ applies to the smallest formula following it, then∧ is to connect the smallest formulas surrounding it, and so on.
Atheory is just a set of formulas and, in this paper, we only consider finite theories. Moreover, ifT is a theory, we use the notationLT to stand for the set of atoms that occur in the theory T, if T = {φ} we denote Lφ. A literal l is either an atom or the negation of an atom.
2.2 Logic systems
We consider a logic simply as a set of formulas that, moreover, satisfies the following two properties: (i) is closed under modus ponens (i.e. if αandα→β are in the logic, then so isβ) and (ii) is closed under substitution (i.e. if a formula αis in the logic, then any other formula obtained by replacing all occurrences of an atombinαwith another formulaβ is still in the logic). The elements of a logic are calledtheoremsand the notation`X αis used to state that the formula αis a theorem ofX (i.e.α∈X). We say that a logicX isweaker than or equal to a logicY ifX ⊆Y, similarly we say thatX isstronger than or equal toY if Y ⊆X.
Hilbert style proof systems There are many different approaches that have been used to specify the meaning of logic formulas or, in other words, to define lo- gics. In Hilbert style proof systems, also known as axiomatic systems, a logic is specified by giving a set of axioms (which is usually assumed to be closed by substitution). This set of axioms specifies, so to speak, the ‘kernel’ of the logic. The actual logic is obtained when this ‘kernel’ is closed with respect to the inference rule of modus ponens. The notation`XF for provability of a logic formulaF in the logicX is usually extended within Hilbert style systems; given a theoryT, we useT `XF to denote the fact that the formulaF can be derived from the axioms of the logic and the formulas contained in T by a sequence
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of applications of modus ponens1. Recall that, in all these definitions, the lo- gic connectives are parameterized by some underlying logic, e.g. the expression
`X (F1∧ · · · ∧Fn)→F actually stands for `X(F1∧X· · · ∧XFn)→XF. Cω logic [6] is defined by the following set of axioms:
Pos1 a→(b→a)
Pos2 (a→(b→c))→((a→b)→(a→c)) Pos3 a∧b→a
Pos4 a∧b→b
Pos5 a→(b→(a∧b)) Pos6 a→(a∨b) Pos7 b→(a∨b)
Pos8 (a→c)→((b→c)→(a∨b→c)) Cω1 a∨ ¬a
Cω2 ¬¬a→a
Note that the first 8 axioms somewhat constraint the meaning of the→,∧and
∨connectives to match our usual intuition. It is a well known result that in any logic satisfying axioms Pos1 and Pos2, and with modus ponens as its unique inference rule, theDeduction Theoremholds [9].
Multivalued logics An alternative way to define the semantics for a logic is by the use of truth values and interpretations. Multivalued logics generalize the idea of using truth tables that are used to determine the validity of formulas in classical logic. The core of a multivalued logic is its domain of values D, where some of such values are special and identified as designated. Logic connectives (e.g. ∧,
∨, →, ¬) are then introduced as operators over D according to the particular definition of the logic.
Aninterpretation is a functionI:L → Dthat maps atoms to elements in the domain. The application ofIis then extended to arbitrary formulas by mapping first the atoms to values in D, and then evaluating the resulting expression in terms of the connectives of the logic (which are defined over D). A formula is said to be atautology if, for every possible interpretation, the formula evaluates to a designated value. The most simple example of a multivalued logic is classical logic where: D ={0,1}, 1 is the unique designated value, and connectives are defined through the usual basic truth tables. IfX is any logic, we write|=X αto denote thatαis a tautology in the logicX. We say thatαis a logical consequence of a set of formulasΓ ={ϕ1, ϕ2, . . . , ϕn} (denoted byΓ |=X α) ifV
Γ →αis a tautology, whereV
Γ stands forϕ1∧ϕ2∧. . .∧ϕn.
Note that in a multivalued logic, so that it can truly be alogic, the implication connective has to satisfy the following property: for every value x∈ D, if there is a designated valuey∈ D such thaty→xis designated, thenxmust also be a designated value. This restriction enforces the validity of modus ponens in the logic. The inference rule of substitution holds without further conditions because of the functional nature of interpretations and how they are evaluated.
Given a theoryT, we define the negation of the theory¬T as{¬F|F∈T}
(the negation symbol is parameterized with respect to some given logic). For
1 We drop the subscriptX in`X when the given logic is clear from the context.
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any pair of theories T and U, we use T `X U to state thatT `X F for every formulaF ∈U.
3 Axiomatization of GLukG
We present the Hilbert-style axiomatization of GLukG that is a slight (equiva- lent) variant of the one presented in [10]. This logic has three primitive logical connectives, namely GL := {→,∧,¬}. GLukG-formulas (or GL-formulas) are formulas built from these connectives in the standard form. We also have three defined connectives:
1. α∨β:= ((α→β)→β)∧((β→α)→α).
2. −α:=α→(¬α∧ ¬¬α).
3. α↔β:= (α→β)∧(β→α).
GLukG Logic has all the axioms of Cωlogic plus the following:
E1 (¬α→ ¬β)↔(¬¬β→ ¬¬α)
E2 ¬¬(α→β)↔((α→β)∧(¬¬α→ ¬¬β)) E3 ¬¬(α∧β)↔(¬¬α∧ ¬¬β)
E4 (β∧ ¬β)→(− −α→α)
Note that Classical logic is obtained from GLukG by adding to the list of axioms any of the following formulas:α→¬¬α, α→(¬α→β),(¬β→¬α)→(α→β).
It is shown in [11]. On the other hand, −α→ ¬αis a theorem in GLukG the “
−” connective is called strong negation. Some experimental work as well as the experience with answer set programming suggested to consider a more useful negation connective for NMR that we will introduce in the following section. See [17] to understand this point in the context of ASP.
Theorem 1 ([11]). For every formula α, α is a tautology in G03 iff α is a theorem in GLukG.
In this paper we consider thestandard substitution, here represented with the usual notation:ϕ[α/p] will denote the formula that results from substituting the formula α for the atom p, wherever it occurs in ϕ. Recall the recursive definition: if ϕ is atomic, then ϕ[α/p] is α when ϕequals p, and ϕ otherwise.
Inductively, ifϕis a formulaϕ1#ϕ2, for any binary connective #. Thenϕ[α/p]
will beϕ1[α/p]#ϕ2[α/p]. Finally, ifϕis a formula of the form¬ϕ1, thenϕ[α/p]
is¬ϕ1[α/p].
3.1 Multivalued logic
It is very important to note that GLukG can also be presented as a multivalued logic. Such presentation is given in [13], where GLukG is calledG03. In this form it is defined through a 3-valued logic with truth values in the domainD={0,1,2}
where 2 is the designated value. The evaluation function of the logic connectives is then defined as follows:x∧y= min(x, y);x∨y= max(x, y); and the¬and→
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connectives are defined according to the truth tables given in Table 1. We write
|=αto denote that the formula αis a tautology, namely thatαevaluates to 2 (the designated value) for every valuation. We say thatαis a logical consequence of a set of formulas Γ ={ϕ1, ϕ2, . . . , ϕn} (denoted byΓ |=α) if VΓ →αis a tautology, whereVΓ stands forϕ1∧ϕ2∧. . .∧ϕn.
In this paper we keep the notionG03to refer to multivalued logic just defined and we use the notion GLukG to refer as the Hilbert system defined at the beginning of this section.
x¬x 0 2 1 2 2 0
→0 1 2 0 2 2 2 1 0 2 2 2 0 1 2
Table 1.Truth tables of connectives in G03.
Theorem 2 (Substitution theorem for G03-logic [10]).Let α,β and ψ be GL-formulas and letpbe an atom. Ifα↔β is a tautology inG03 thenψ[α/p]↔ ψ[β/p]is a tautology in G03.
Corollary 1 ([10]). Let α, β andψ beGL-formulas and let p be an atom. If α↔β is a theorem in GLukGthenψ[α/p]↔ψ[β/p]is a theorem in GLukG.
4 Main Results
We presentN50, a 5-valued logic. We will use the set of values{−2,−1,0,1,2}.
Valid formulas evaluate to 2. The connectives ∧ and ∨correspond to the min and maxfunctions in the usual way. For the other connectives, the associated truth tables are as follows:
→ −2 −1 0 1 2
−2 2 2 2 2 2
−1 2 2 2 2 2
0 2 2 2 2 2
1−1 −1 0 2 2 2−2 −1 0 1 2
¬
−2 2
−1 2 0 2 1 2 2−2
∼
−2 2
−1 1 0 0 1−1 2−2
We have defined 5 logical connectives, namely Nc := {→,∧,∨,¬,∼}. N- formulas are built from these set of connectives. If α always evaluates to the designated value then it is called a tautology.
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Remark 1. Observed the following:
1.|=∼(α→β)↔α∧ ∼β. 7.− |=¬ ∼α→((¬ ∼α→α)→α).
2.|=∼(α∧β)↔ ∼α∨ ∼β. 8.− |=∼α→((¬ ∼α→α)→ ¬α).
3.|=∼(α∨β)↔ ∼α∧ ∼β. 9.−2∼α→((¬ ∼α→α)→α).
4.|=α↔ ∼∼α. 10.−2∼α→((¬ ∼α→α)→ ¬ ∼α).
5.|=∼ ¬α↔ ¬¬α. 11.−2¬α→((¬ ∼α→α)→ ¬ ∼α).
6.|=∼α→ ¬α. 12.−2¬α→((¬ ∼α→α)→ ∼α).
What is important about this remark is that these formulae have the same structure than those theorems of Nelson introduced previously in [3] for the construction of extensions, with strong negation, of intuitionistic logic; In fact, formulae 1−6 show the classical-like behavior of strong negation, in particular, formula 6 honors the adjectivestrong.
Theorem 3. Given αandβ two formulae, then:
1. If αandα→β are tautologies, thenβ is also a tautology.
2. If αis a tautology, then¬¬αis also a tautology.
Proof. The proof of 1 is straightforward; If αand α→β are tautologies, then they always evaluate to the designated value and, looking at the implication table given for N50, we see that (since αevaluates to 2) there is only one case in which α→β evaluates to 2, and it is precisely whenβ evaluates to 2. This concludes the proof of 1.
To prove 2, we see that ifαevaluates to 2, then¬¬αalso evaluates to 2.
Note:N50 logic is in some way similar toN5logic (see [3] for more information onN5). The only difference is that inN5,¬1 =−1, but inN50,¬1 = 2. Moreover, withN50 logic we can expressN5 logic.
Theorem 4. N50 logic can expressN5 logic.
Proof. It suffices to see that we can express the constant⊥ofN5by the formula
¬a∧ ¬¬a of N50, where a is any formula. Hence, we can also express the ¬α formula ofN5 byα→ ⊥in N50.
Lemma 1. LetIbe an interpretation (based onG03). LetJI be an interpretation (based on N50) defined for every atoma as follows:
JI(a) =
I(a) if I(a)>0
−2 otherwise Then, for everyGL-formulaα,
JI(α) =
I(α) if I(α)>0
<0 otherwise
Proof. The proof is by induction on the numbernof ocurrences of→,∧,∨,¬,∼:
ifn= 0, αis just an atoma, the proof is followed by definition ofJI. Assume now that lemma holds for all j≤n.
Case 1.αis¬β. Then β has fewer thannconnectives.
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1a. 0< I(α) thenI(α) = 2, henceI(β) = 1 orI(β) = 0. IfI(β) = 1, by inductive hypothesis, we have thatJI(β) =I(β) = 1 soJI(α) =JI(¬β) = 2 =I(α).
1b.I(β) = 0, this subcase is analogous to subcase 1a.
Case 2.αisβ→γ. Thenβ andγ have fewer thannconnectives.
2a. 0< I(α) thenI(α) = 1 orI(α) = 2.
i) Assume 1 =I(α), soI(β) = 2 andI(γ) = 1. By inductive hypothesis,JI(β) = I(β) = 2 andJI(γ) =I(γ) = 1 thusJI(α) = 1 =I(α).
ii) IfI(α) = 2 is worked analogously the previous subcase.
2b. I(α) = 0, we have by inductive hypothesis 0 < I(β) = JI(β) and I(γ) = JI(γ)<0, soJI(α)<0.
Case 3.αisβ∧γ. Thenβ andγ have fewer thannconnectives.
3a. 0 < I(α) then by inductive hypothesis 0 < I(α) = min{I(β), I(γ)} = min{JI(β), JI(γ)}=JI(α).
3b.I(α)≤0 is similar to subcase 3a.
Finally, the caseαisβ∨γis proved analogously.
A very important result is thatN50 logic is a conservative extension of G03 logic, as the following theorem shows.
Theorem 5. For everyGL-formulaα,αis a tautology inN50 iffαis a tautology inG03.
Proof. Suppose thatαis a tautology inG03, then by Theorem 1,αis a theorem in GLukG. We proceed by induction on the length of the proof ofα. The base case is immediate from the fact that all GLukG axioms are tautologies in N50. We suppose that any proposition, that has a proof with at mostnsteps, satisfies the induction hypothesis.
Let α be a proposition whose proof requires exactly n steps and let be B1, . . . , Bn, a proof of α. We are done if Bn = α is a GLukG axiom. If α is a Modus Ponens consequence, then by Theorem 3, if β andβ → αare tauto- logies, thenαis tautology. This observation and the inductive hypothesis finish the proof.
For the other implication we suppose thatαis not a tautology in G03, then there exists an interpretation I (based on G03) such that I(α)6= 2. By Lemma 1, there existsJI, an interpretation based onN50, such that JI(α)6= 2. Hence,α is not a tautology ofN50, as desired.
4.1 Substitution
A particular feature of our N50 logic is that the symbol ↔ does not define a congruential relation on formulas, note that it can be the case thatα↔β is a tautology, but ∼α↔ ∼β doesn’t. A particular example is the following: Take α1 to be ∼ (a→b) and α2 to be a∧ ∼ b. Clearly α1↔α2 is a tautology, but
∼α1↔ ∼α2does not (take I(a) =I(b) = 1). This property also holds inN5. Thus, when we refer to equivalence of formulas, we will have to be more precise and make some particular considerations. The term weak equivalence will mean that α↔β is a tautology. There is a stronger notion of equivalence ofN-formulas, which we will callN50-equivalence, and it holds when bothα↔β
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and∼α↔ ∼β are tautologies. For this purpose, we define a new connective⇔.
We write α⇔β to denote the formula: (α↔β)∧(∼α↔ ∼β). The reader can easily verify thatα↔β is a tautology iff for every valuationv,v(α)>0 implies v(α) =v(β), while α⇔β is a tautology iff for every valuation v, v(α) =v(β).
This can be seen in the following truth tables:
↔-2 -1 0 1 2 -2 2 2 2 -1 -2 -1 2 2 2 -1 -1 0 2 2 2 0 0 1 -1 -1 0 2 1 2 -2 -1 0 1 2
⇔-2 -1 0 1 2 -2 2 1 0 -1 -2 -1 1 2 0 -1 -1 0 0 0 2 0 0 1 -1 -1 0 2 1 2 -2 -1 0 1 2
Theorem 6 (Basic Substitution theorem). Let α,β andψ be N-formulas and letpbe an atom. Ifα⇔βis a tautology thenψ[α/p]↔ψ[β/p]is a tautology.
Proof. Ifα⇔β is tautology then for everyv, anN-valuation,v(α) =v(β) (see the comments at the end of the previous paragraph). Therefore, v(ψ[α/p]) = ψ[β/p]) so,v(ψ[α/p]↔ψ[β/p]) = 2.
To be able to apply standard substitution we requireN50-equivalence of for- mulas to hold. However, in certain cases this condition may be too strong. We are also interested in the particular cases where weak equivalence of formulas suffices for substituting. The first such case is when substitution is not done inside the scope of a∼symbol.
Lemma 2. Let α, β and ψ be N-formulas and let p be an atom such that p does not occur inψ within the scope of a∼symbol. Ifα↔β is a tautology then ψ[α/p]↔ψ[β/p]is a tautology.
Proof. It is of high importance to remember (see the comments at the end of the first paragraph) that:θ↔η is a tautology iff for every valuationv,v(θ)>0 impliesv(θ) =v(η). By structural induction:
Base case:
Ifψ=q,qan atom, we have:
ψ[α/p]↔ψ[β/p] =
(q↔qifp6=q
α↔β ifp=q,(hypothesis) therefore it is a tautology.
We suppose that the inductive case is satisfied:
Ifψ=¬ϕ
By inductive hypothesis, for everyN05-valuation,ϕ[α/p]↔ϕ[β/p] is a tauto- logy. If 0< v(ϕ[α/p]) thenv(¬ϕ[α/p]) =v(¬ϕ[β/p]), thereforeψ[α/p]↔ψ[β/p]
is a tautology.
Remark 2. Ifv(ϕ1)≤0 andv(ϕ2)≤0 impliesv(ϕ1→ϕ2) = 2
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Ifψ=ϕ1∧ϕ2
Suppose that 0< v((ϕ1∧ϕ2)[α/p]) then, 0<min{v(ϕi[α/p])}i=1,2.Hence, by the inductive hypothesisv(ψi[α/p]) =v(ψi[β/p]) fori= 1,2. Thenv(ψ[α/p]) = min{v(ϕi[α/p])}i=1,2 = min{v(ϕi[β/p])}i=1,2=v(ψ[β/p]). Therefore ψ[α/p]↔ ψ[β/p] is a tautology.
ψ=ϕ1→ϕ2
Remember that by inductive hypothesis we have ϕ1[α/p] ↔ ϕ1[β/p] and ϕ2[α/p]↔ϕ2[β/p] are tautologies.
Suppose that 0< v(ψ[α/p]) =v(ϕ1[α/p]→ϕ2[α/p]).
We have two cases in which we can apply the inductive hypothesis:
1. v(ϕ1[α/p])>0 then by inductive hypothesis, we havev(ϕ1[α/p]) =v(ϕ1[β/p]), then 0< v(ϕ2[α/p]), again by inductive hypothesisv(ϕ2[α/p]) =v(ϕ2[β/p]), thereforev(ϕ1[α/p]→ϕ2[α/p]) =v(ϕ1[β/p]→ϕ2[β/p]).
2. v(ϕ2[α/p]) > 0 then v(ϕ2[α/p]) = v(ϕ2[β/p]). Since ϕ1[α/p] ↔ ϕ1[β/p]
and ϕ2[α/p] ↔ ϕ2[β/p] are tautologies, then we have that both ϕ1[α/p]
andϕ1[β/p] are negative or null (not necessarily in the same case). Finally v(ϕ1[α/p]→ϕ2[α/p]) =v(ϕ1[β/p]→ϕ2[β/p]).
4.2 Standard form
We present the notion of a standard form of a formula.
Definition 1. We define the functionS:N-formulae→N-formulae as follows:
If ais an atom andϕis an N-formula, then S(a) =a,
S(¬a) =¬a, S(∼a) =∼a, S(∼ ¬α) =S(¬¬α), S(¬α) =¬S(α),
S(α→β) =S(α)→S(β), S(α∧β) =S(α)∧S(β), S(∼(α→β)) =S(α)∧S(∼β), S(∼(α∧β)) =S(∼α)∨S(∼β), S(∼∼α) =S(α).
Definition 2 (Standard Form). An N-formula ϕ is said to be in standard form ifS(ϕ) =ϕ
Intuitively a formula is in standard form if it has all occurrences of the∼ connective just in front of an atom.
Example 1. Take the formula ϕ:=∼(a→ ¬b)∧ ∼c. Then its standard form is S(ϕ) :=a∧b∧ ∼c.
Lemma 3. For anyN-formulaϕ,ϕis a tautology inN50 iffS(ϕ)is a tautology inN50.
Proof. By structural induction:
Base case: It is vacuously true.
We suppose that the inductive case is satisfied:
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ψ=¬ϕis a tautology iffϕevaluates to−2,−1,0 or 1 iff, by inductive hypothe- sis, 2 =¬S(ϕ) =S(ψ).
ψ = ϕ1∧ϕ2 is a tautology iff ϕ1 and ϕ2 have the same value, by inductive hypothesis,S(ψ) =S(ϕ1)∧S(ϕ2) is tautology.
Ifψ=ϕ1→ϕ2is tautology, we have two cases:
1.- if ϕ1 has any of the values −2,−1 or 0 then by inductive hypothesis also S(ϕ1) the same value; hence it does not matter the value of value S(ϕ2) and S(ψ) =S(ϕ1→ϕ2) =S(ϕ1)→S(ϕ2) is a tautology, and reciprocally
2.- if the value ofϕ1 is 1 or 2 then ϕ2 has the same value, respectively; hence, by the inductive hypothesis we haveS(ψ) is tautology.
Ifψ=∼ϕis a tautology then by remark 1 and the inductive hypothesis we have that S(ψ) is a tautology.
Finally, if ψ =∼ ¬ϕis a tautology, then S(ψ) =S(∼ ¬ϕ) =S(¬¬ϕ) =S(ϕ), but∼ ¬ϕ→ϕis a tautology, thenϕis tautology, hence by inductive hypothesis S(ϕ) is a tautology.
5 Conclusions and Future Work
We introduce a family of paraconsistent logics extended by a strong paracon- sistent negation operator. We study one particular logic, that we call N50, and we show that it is sound with respect to N-GLukG. For future work one can consider the formal construction of non-monotonic semantics based onN50. This seems to be an easy task thanks to the experience of the construction of the answer set semantics based onN5 logic.
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